=Paper=
{{Paper
|id=Vol-2500/paper_9
|storemode=property
|title=Optimizing Classification Thresholds of Status of Transionospheric Communication Channel
Distributed
According to Rayleigh Distibution Law for Decreased Quadrocopter’s Positioning Errors
|pdfUrl=https://ceur-ws.org/Vol-2500/paper_9.pdf
|volume=Vol-2500
|authors=Gennadiy I. Linets,Sergey V. Melnikov,Michael A. Isaev,Alexander M. Isaev
}}
==Optimizing Classification Thresholds of Status of Transionospheric Communication Channel
Distributed
According to Rayleigh Distibution Law for Decreased Quadrocopter’s Positioning Errors
==
https://ceur-ws.org/Vol-2500/paper_9.pdf
Optimizing classification thresholds of status of
transionospheric communication channel distributed
according to rayleigh distibution law for decreased
quadrocopter’s positioning errors
Gennadiy I. Linets Sergey V. Melnikov
North-Caucasus Federal University North-Caucasus Federal University
Stavropol, Russian Federation Stavropol, Russian Federation
kbytw@mail.ru territoreer@yandex.ru
Michael A. Isaev Alexander M. Isaev
North-Caucasus Federal University North-Caucasus Federal University
Stavropol, Russian Federation Stavropol, Russian Federation
mrraptor26@gmail.com isaev@stilsoft.ru
Abstract
Unmanned aerial vehicle (UAV), today, plays prominent role in many
parts of human activity. One of serious problems exploitation UAV
is accuracy of positioning. Satellite clocks, satellite orbits, ionosphere
and troposphere, multipath, etc - all of this are sources of signal’s fluc-
tuation. But largest fluctuation of signal is created by ionosphera. One
of distribution laws (Nakagami, normal distribution law, Rayleigh and
Rice - most common situations) characterize Ionosphere’s fluctuations.
For solve this problem supposed to using of adapting control system,
based on Flight Control Center. At working this system there exist
probability of making type I and type II errors during salvation of iden-
tification problems. This work proposes an approach to implementing
the threshold optimization for classifying the states of satellite com-
munication systems, when disorting of signals described by Rayleigh
distribution law.
1 Introduction
When receiving signals in satellite communication channels there frequently appears a fluctuation of the signal
amplitude in time. The fluctuations usually take place as a random process with a qausiperiod from fractions of
a second to dozens of minutes, and their main descriptive feature is fading depth. The fading depth is defined
by the deviation of the instantaneous amplitude of the signal from the set level (as a rule, median one) and can
reach dozens of decibels. The level of the fluctuating signal can be estimated with the statistical method. In
Copyright 2019 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
In: S. Hölldobler, A. Malikov (eds.): Proceedings of the YSIP-3 Workshop, Stavropol and Arkhyz, Russian Federation,
17-09-2019–20-09-2019, published at http://ceur-ws.org
1
general, the nonstationary fading process is usually divided into two stationary processes: with fluctuations of
the medium field values and fast fluctuations around medium values. The first type of fluctuations refers to slow
fading, whereas the second type refers to the fast one [Taa16].
The most common distribution of the signal amplitude at fast interference fading is close to Rayleigh distri-
bution.
The energy loss in a satellite communication channel, when there is Rayleigh fading, turns out to be quite
substantial [Pin19]. Therefore, it is necessary to provide the probability of the wrong data reception, in order to
keep the quality of communication in case of a Rayleigh fading not less than 10-5.
One of the methods to keep the set probability of the wrong data receipt at the necessary level is to use
an automated quality control and management system, which will assess the state of the transitionspheric
communication channels in the mode close to real time. Based on the transitionspheric communication channels
state assessment, it is possible to identify the required control action which will allow to keep the necessary
probability of the wrong reception of a GPS/GLONASS signal, and thereby to provide the required accuracy
of remotely piloted aircraft system positioning. Thus, it is necessary to solve the task of the transionspheric
communication channels state identification at Rayleigh fading in order to develop the appropriate control actions
leading to the required remotely piloted aircraft system positioning accuracy.
During the transionspheric communication channels state identification, errors of the first and second kind
may appear, which may lead to an inappropriate control action or to an untimely reaction to the change in
the transionspheric communication channels state, resulting in the deterioration of the remotely piloted aircraft
system positioning accuracy.
2 Analysis
Let us analyze the factors that influence the remotely piloted aircraft system positioning accuracy. The main
error types and the inaccuracies in the remotely piloted aircraft system positioning caused by them are given in
Table 1 [Ste12].
Table 1: The main error types and the inaccuracies in the remotely piloted aircraft system positioning
Error type Caused inaccuracy (m)
Ephemeris data from 1 to 5
Satellite clock Up to 3
Ionosphere influence From 2 to 50
Troposphere effect Around 1
Signal reflection Around 1
Receiver measurements Up to 0.5
It is obvious that ionosphere has the most influence on the remotely piloted aircraft system positioning
accuracy. Satellite communication channels are largely influenced by ionosphere due to the electron density and
its changes, which leads to high level Rayleigh fading in satellite radio-signals. The ratio signal/noise (S/N)
experiences fading depending on the changes in electron density.
If the fading level in a communication channel is rather high, i.e. the number of square components is big, and
none of them exceeds in its amplitude all the rest, then the amplitude signal distribution is defined by Rayleigh
distribution, which is described by [Lam10]:
r r2
pR (r) = exp(− )
σ2 2σ 2
where r is the received signal amplitude; 2σ is the average signal energy.
The most common error metric in satellite communication channels with character oriented messaging without
antinoise coding is error probability per message bit. Modern requirements of the International Communication
Union set limits for this parameter in the required in practice value not less, than 10−5 [Ion16]. The error
probability is defined [Ion16]:
k12 k 2 + h2b0
Perr = 0.5 · · exp(− )
k12 + h2b0 k12 + h2b0
2
where h2b0 is a normalized ratio signal/noise at receiver input; k - is the ratio of effective voltage of the regular
and diffuse signal components at the receiver input, where k12 = 1 + k2.
When the regular component is absent at the receiver input (scattering in the communication channel), for a
rather little error probability and high hb0 values, the probability of the wrong receipt equals [Ion16]:
k12
Perr = 0.5 · · exp(−k 2 )
h2b0
With typical for practice values of k = 57dB [Ion16], the losses in the Rayleigh channel are 7-14 dB higher than
in channel with Rician fading, which is a considerable value. Antinoise coding, implemented in channels with
permanent parameters, at slow fading provides an advantage of several dB [Ion16], which does not provide a
compensation of the antinoise loss. An automatic control and management system is to carry out the controlled
object condition identification, to work out the control action (based on the management goals and taking into
account the current condition of the environment), to realize the control action in the automated mode close to
real time. This will provide a timely reaction to the changes in the condition of transionospheric communication
channels characterized by high uncertainty. When identifying the controlled object condition there appear errors
of the first and the second kind. In statistical hypothesis testing a type I error is the rejection of a true null
hypothesis (also known as a ”false positive” finding or conclusion), while a type II error is the non-rejection of
a false null hypothesis (also known as a ”false negative” finding or conclusion). It is necessary to determine the
optimal value of the classification threshold as the error data directly influence the object positioning accuracy.
The article suggests a solution for the optimization of the thresholds of the first and second type error classification
for an automated system of control and management of transionospheric communication channels, the condition
of which is described by the Rayleigh distribution law.
3 Problem Statement
In general, the task aimed to identify objects could be reduced to the verification of numerous hypotheses
H1 , H2 , . . . , Hi , . . . , Hn , where Hi is a hypothesis implying the objects belonging to Class Ai . Lets assume that
the a priori distributions of these hypotheses probabilities are set. It is known whats the likelihood the object
n
P
P (Hi ) can belong to class Ai (or how often the object of this class is appeared). Moreover, P (Hi ) = 1, as
i=1
the object is to be pertained to a certain class. The conditional density of distribution is pi (x) = p(xi /Hi ).
Two hypotheses H1 = N and H2 = N̄ are used in the identification system under design at corresponding
to them a priori probabilities of situation, emerging in the network as a normal p1 = p(H1 ) = p(N ) one and an
abnormal p2 = p(H2 ) = p(N̄ ) . And also p1 + p2 = 1.
It is required to find a decision rule ensuring the top accuracy in the identification system. Using the Neyman-
Pearson criterion we fix the probability of ”false alarm” Pf.a. at stable level C and claim the minimum of pass
min
error Ppass of TN operating trouble. Then
n
Y
min
Ppass = p2 [1 − β̄(xoi )] (1)
i=1
n
Q
at restriction of Pf.a. = p1 α(xoi ) = C, const, where α(x0 ) are Type I errors; and β(x0 ) are Type II errors.
i=1
4 Solution
During operation of transionospheric communication channels the status of which could be described by a large
number of parameters, performance monitoring of their working efficiency is to be carried out using stage-by-stage
principle of classification.
At the first stage, the transionospheric communication channels status is checked by the summarized index
and if the abnormality is detected, a stricter control is carried out at the second and next stages on which its
real status is defined.
At each of the stages the control system makes Type I α(xo i) and Type II β(xo i) errors. Type I and II errors
3
(”false alarm” and ”abnormal situation” errors), are defined as follows:
Z∞
α(xoi ) = f (xi /N )dxi (2)
xoi
Zxoi
β(xoi ) = f (xi /N̄ )dxi (3)
−∞
The activity of the monitoring error evaluation system can be described by a probability graph.
The transionospheric communication channels are characterized with the certain states of N and N̄ , under
which we will mean normal and abnormal operation in the process of its functioning which (in the monitoring
system) are respectively displayed into the normal A and abnormal¯state.
A priori probabilities of the normal and abnormal states in transionospheric communication channels are
detected correspondingly by a priori probabilities of the p1 and p2 states. For all the stages, with abnormal
situation omission, were obtained resultant errors:
n
Y
Ppass = p2 β1 + p2 β̄1 β2 + · · · + p2 β̄1 β̄2 . . . βn = p2 [1 − β̄(xoi )] (4)
i=1
and ”false alarm” errors:
n
Y
Pf.a = p1 α1 α2 . . . αn = p1 α(xoi ) (5)
i=1
where:β̄i = β̄(xoi ), αi = α(xoi ), p1 = 1 − p2 a priori probability of a normal situation occurrence; p2 a priori
probability of its absence.
The problem of thresholding xoi for each stage is currently central. Lets make tradeoffs of thresholds. Following
the Neyman-Pearson criterion we will set the probability of false alarm at certain given C level. Then, for the
entire network, we get
n
Y
Pf.a = p1 α(xoi ) = C (6)
i=1
Having minimized the probability of abnormal situation omission we get
n
Y
min
Ppass = min p2 [1 − β̄(xoi )] (7)
xoi
i=1
As a result, the decision rule ensuring the highest level of accuracy for the identification system is the Neyman-
Pearson criterion which is detected for the TN by expressions (6) and (7). The minimization problem of the
function (7), where variables xoi are linked by functional dependence (6) is the constrained optimization problem.
Lets form a functional of optimization
n
Y n
Y
F = p2 [1 − β̄(xoi )] + λp1 α(xoi ) (8)
i=1 i=1
where λ is an undetermined Lagrange multiplier.
dF
Calculating the derivatives dx oi
= 0 we get a system of n equations:
dβ̄(xo1 ) dα(xo1 )
p2 β̄(xo2 ) . . . β̄(xon ) − λp1 α(xo2 ) . . . α(xon ) = 0;
dxo1 dxo1
... (9)
dβ̄(xon ) dα(xon )
p2 β̄(xo1 ) . . . β̄(xo(n−1) ) + λp1 α(xo1 ) . . . α(xo(n−1) ) = 0;
dxon dxon
4
which, along with equation (6) allow to find the undetermined Lagrange multiplier λ and n of xoi variables.
Since the system (9) contains n of xoi variables and undetermined Lagrange multiplier λ, which must be
defined, while the equation system (9) has n equations and n + 1 of variables, then for finding its unambiguous
solution one more equation should be added, in the capacity of which we take the tolerance probability equation
for ”false alarm” (6).
Equations (9) and (6) including (2) and (3) after differentiation on the lower and higher threshold take the
form:
xRo2 xRon
f (x2 /N̄ )dx2 . . . f (xn /N̄ )dxn
−∞ −∞ p1 f (xo1 /N )
R∞ R∞ =λ ;
p2 f (xo1 /N̄ )
f (x2 /N )dx2 . . . f (xn /N )dxn
xo2 xon
... (10)
xRo1 xo(n−1)
R
f (x1 /N̄ )dx1 . . . f (xn−1 /N̄ )dxn−1
−∞ −∞ p1 f (xon /N )
R∞ R∞ =λ ;
p2 f (xon /N̄ )
f (x1 /N )dx1 . . . f (xn−1 /N )dxn−1
xo1 xo(n−1)
n Z ∞
Y
p1 f (xi /N )dxi = C.
i=1x
oi
Under certain distribution laws and f (xi /N̄ ) this problem has a single value solution. In equations (10)
the unknowns are optimal thresholds for classification (x∗o1 , x∗o2 , . . . , x∗oi , . . . , x∗on ) at each of the stages, offering
a minimum of the function (7), i.e. the minimal probability of abnormal situation omission. The solution
clearly allows defining the probability of transionospheric communication channels operating trouble absence
(probability of normal operation):
min
P̄pass = 1 − Ppass (11)
At each of the following stages, the information about correct solution of β̄i = 1 − βi (information about
normal transionospheric communication channels functioning) is analyzed. When the Type I or Type II errors
are detected, their specification is carried out by means of the following stage procedure. Let us define the
procedure of error detection by the feature x as the first stage of two-stage procedure, while for the second stage
we will detect the error by the feature y. Generally, the identification problem solution is determined by Type I
and Type II errors. Now lets write down the functions for distribution density of the feature x at transionospheric
communication channels problem-free functioning f (x/I = f1 (x)) and at trouble functioning f (x/I¯ = f1 (x)).
Then the errors of Type I and Type II of the detector (stage 1) are :
Z∞ Zxo
αo = f1 (x)dx βo = f2 (x)dx (12)
xo −∞
Type II errors for the recognizer (Stage 2) are defined:
Z∞ Zyo
αp = f1 (y)dy βp = f2 (y)dy (13)
yo −∞
Transionospheric communication channels has the following conditions: ”1” - the system is out of order, the
failure was not detected; ”2” the system is operational, it was found as workable; ”3” - failure was detected
and recognized (abnormal situation presence); ”4” the system is workable, false detection and recognition (false
alarm); ”5” system is out of order, failure was detected but not recognized (abnormal situation omission); ”6”
- the system is in order, a false detection and correct recognition.
5
In view of formulae (12) and (13), formulae (6) and (7) take the form:
Z∞ Z∞
Pf.a. = p1 f1 (x)dx f1 (y)dy = C = const (14)
xo yo
Z∞ Z∞
min
Ppass = minp2 (1 − f2 (x)dx f2 (y)dy) (15)
xo yo
Since in the expression (14) the thresholds xo and yo are linked with one functional dependence xo = φ(yo ),
having differentiated (15) by yo and set it to zero, we get:
Z∞ Z∞
dxo
· f2 (xo ) · f2 (y)dy + f2 (yo ) · f2 (x)dx = 0 (16)
dyo
yo xo
At free-hand laws of x and y features distribution, particularly, at normal law, there is no possibility to obtain
an exact solution of the classification thresholds optimization problem. However, in certain cases, at Rayleigh
distribution laws, in particular, the solution could be obtained in its final form.
Lets set densities of features x and y distribution probability in the form of Rayleigh distribution laws. The
Type I and Type II errors are in the form of:
Z∞ Zxo Z∞ Zyo
αo = f1 (x)dx; βo = f2 (x)dx; αp = f1 (y)dy; βp = f2 (y)dy;
xo a yo b
2 (x−a)2 2 (y−b)2
− x2 − − y2
f1 (x) = xe ; f2 (x) = (x − a)e 2 ; f1 (y) = ye ; f2 (y) = (y − b)e− 2 ;
Equations (14) and (16) are transformed into expressions:
p1
x2o + yo2 = 2ln (17)
C
dxo
· (xo − a) + (yo − b) = 0 (18)
dyo
Having differentiated (17) by o by substituting the result in (18), we obtain optimal classification thresholds:
s s
2lnp1 /C ∗ 2lnp1 /C
x∗o = ; yo = (19)
1 + ( ab )2 1 + ( ab )2
5 Example
Lets analyze increasing UAV positioning errors, using program u-center. Analized data was got in real experiment
from UAV.
Fig. 1 demonstrate example of UAV positioning errors distribution, when navigation area is fluctuating.
Fig. 1.left demonstrating example of UAV positioning errors distribution for clear sky, i.e. normal state of
navigation area. UAV positioning errors less than 0,9 m. That data was get with PDOP (Position dilution of
precision) = 1, signal/noise ratio in measuring channels - not less than 45 dBm / Hz, the time delay of the signal
is 1 . . . 2 ns.
Fig 1.right demonstrating example of UAV positioning errors distribution caused by ionosphere fluctuation.
UAV positioning errors more than 7 m. That data was get with PDOP = 1 . . . 2, signal/noise ratio in measuring
channels 33 . . . 28 dBm / Hz, the time delay of the signal is 2 . . . 5 ns.
6
Figure 1: Positioning error increase of UAV into ionospheric fluctuations area.
Figure 2: Dependence curves: a) optimal threshold of detector from optimal threshold of recognizer; b) optimal
threshold of detector from a priori probability of networks normal status.
Thereby, when its fluctuation of navigation area, we must establish the fact of fluctuating and, after that,
identify state of transionospheric communication channels for develop necessary control procedure. This will
allow to hold signal/noise of GPS/GLONASS in required values. p
Dependences in Fig. 2a, built according to the formula (16) are circumferences of the 2lnp1 /C radius.
With the increase of a priori probability of normal connection status p1 , the radius is increased subject to the
logarithmic law.
6 Conclusions
The problem of optimizing classification thresholds of the first and second kind of errors of trans-ionospheric
communication channels control and monitoring system, described by the Rayleigh distribution law of random
variables is solved in this paper. The authors have obtained the common solution of problem the optimal
classification thresholds for the anomalous situation detector and recognizer. In future work, it is advisable to
conduct an investigation on the dependencies of the communication channel states for the distribution law of
random variables described by Nakagami distribution functions.
6.0.1 Acknowledgements
This research is supported by the scientific project Development of a multi-rotor robotic unmanned aerial ve-
hicle using a strapdown inertial navigation system in the Federal Target Program 2014-2020 (unique identifier
RFMEFI57818X0222) with the financial support of the Ministry of Science and Higher Education of Russian
Federation.
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